Question

A 5 digit number divisible by 3 is to be formed using the numeral 0 1 2 3 4 5 without repetition the total number of ways this can be done is

Answer 1

firstly ur question is wrong
your question is
**five-digit number divisible by 3 is to be formed using numerical 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways this can be done is:

We should determine which 5 digits from given 6, would form the 5 digit number divisible by 3.

We have six digits: 0, 1, 2, 3, 4, 5. Their sum=15.

For a number to be divisible by 3 the sum of the digits must be divisible by 3. As the sum of the six given numbers is 15 (divisible by 3) only 5 digits good to form our 5 digit number would be 15-0={1, 2, 3, 4, 5} and 15-3={0, 1, 2, 4, 5}. Meaning that no other 5 from given six will total the number divisible by 3.

Second step:

We have two set of numbers:
1, 2, 3, 4, 5 and 0, 1, 2, 4, 5. How many 5 digit numbers can be formed using these two sets:

1, 2, 3, 4, 5 --> 5! as any combination of these digits would give us 5 digit number divisible by 3. 5!=120.

0, 1, 2, 4, 5 --> here we can not use 0 as the first digit, otherwise number won't be any more 5 digit and become 4 digit. So, desired # would be total combinations 5!, minus combinations
with 0 as the first digit (combination of 4) 4! --> 5!-4!=4!(5-1)=4!*4=96

120+96=216

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Answer 2

Answer:

The sum of the digits of any number that is divisible by 3 is also divisible by 3.

Now, we are given with six digits i.e., 0, 1, 2, 3, 4 and 5 and the sum of all the six digits is 15.

Thus, we should not use either 0 or 3 while forming the five digit numbers.

Case 1 : If we do not use 0

Then the remaining 5 digits can be arranged in 5! ways = 120 numbers.

Case 2 : If we do not use 3

Then the arrangements should take into account that 0 cannot be the first digit

⇒ The first digit from the left can be any of the 4 digits 1, 2, 4 or 5.

Then the remaining 4 digits including 0 can be arranged in the other 4 places in 4! ways.

So, there will be 4 × 4! = 96 numbers.

Hence, the total number of 5 digit numbers divisible by 3 that can be formed using the digits 0 to 5 = 120 + 96 = 216 .

Step-by-step explanation: